Fubini's theorem on differentiation

Fubini's theorem on differentiation In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.[1] Statement Assume {displaystyle Isubseteq mathbb {R} } is an interval and that for every natural number k, {displaystyle f_{k}:Ito mathbb {R} } is an increasing function. If, {displaystyle s(x):=sum _{k=1}^{infty }f_{k}(x)} exists for all {displaystyle xin I,} then, {displaystyle s'(x)=sum _{k=1}^{infty }f_{k}'(x)} almost everywhere in I.[1] In general, if we don't suppose fk is increasing for every k, in order to get the same conclusion, we need a stricter condition like uniform convergence of {displaystyle sum _{k=1}^{n}f_{k}'(x)} on I for every n.[2] References ^ Jump up to: a b Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529. ^ Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill, p. 152. Categories: Theorems in real analysisTheorems in measure theory